related by the Dold-Kan correspondence
category with duals (list of them)
dualizable object (what they have)
is itself a cofibration, which, furthermore, is acyclic if or is.
(unit axiom) For every cofibrant object and every cofibrant resolution of the tensor unit , the resulting morphism
is a weak equivalence.
In particular if the tensor unit happens to be cofibrant, then the unit axiom in def. 1 is implied by the pushout-product axiom. (Because then is a weak equivalence between cofibrant objects and such are preserved by functors that preserve acyclic cofibrations, by Ken Brown's lemma. )
We say a monoidal model category, def. 1, satisfies the monoid axiom, def. 1, if every morphism that is obtained as a transfinite composition of pushouts of tensor products of acyclic cofibrations with any object is a weak equivalence.
In particular, the axiom in def. 2 says that for every object the functor sends acyclic cofibrations to weak equivalences.
Consider the explicit model of as the category of fibrant-cofibrant objects in with left/right-homotopy classes of morphisms between them (discussed at homotopy category of a model category).
A derived functor exists if its restriction to this subcategory preserves weak equivalences. Now the pushout-product axiom implies that on the subcategory of cofibrant objects the functor preserves acyclic cofibrations, and then the preservation of all weak equivalences follows by Ken Brown's lemma.
To that end, consider the construction of the localization functor via a fixed but arbitrary choice of (co-)fibrant replacements and , assumed to be the identity on (co-)fibrant objects. We fix notation as follows:
all morphisms are weak equivalences: For the first two this is due to the pushout-product axiom, for the third this is due to the unit axiom on a monoidal model category. It follows that under this zig-zig gives an isomorphism
and similarly for tensoring with from the right.
To exhibit lax monoidal structure on , we need to construct a natural transformation
By the definitions at homotopy category of a model category, the morphism in question is to be of the form
To this end, consider the zig-zag
and observe that the two morphisms on the left are weak equivalences, as indicated, by the pushout-product axiom satisfied by .
Hence applying to this zig-zag, which is given by the two horizontal part of the following digram
and inverting the first two morphisms, this yields a natural transformation as required.
To see that this satisfies associativity if the monoid axiom holds, tensor the entire diagram above on the right with and consider the following pasting composite:
Observe that under the total top zig-zag in this diagram gives
Now by the monoid axiom (but not by the pushout-product axiom!), the horizontal maps in the square in the bottom right (labeled ) are weak equivalences. This implies that the total horizontal part of the diagram is a zig-zag in the first place, and that under the total top zig-zag is equal to the image of that total bottom zig-zag. But by functoriality of , that image of the bottom zig-zag is
The same argument applies to left tensoring with instead of right tensoring, and so in both cases we reduce to the same morphism in the homotopy category, thus showing the associativity condition on the transformation that exhibits as a lax monoidal functor.
The classical model structure on pointed topological spaces or pointed simplicial sets with the smash product of pointed objects.
With respect to a symmetric monoidal smash product of spectra:
The standard example of a monoidal model category whose unit is not cofibrant is the category of EKMM S-modules.
Let be a category equipped with the structure of
the model structure is cofibrantly generated;
the tensor unit is cofibrant.
preserves and reflects fibrations and weak equivalences.
See for instance (BergerMoerdijk 2.5).
A general standard reference is
The monoidal structures for a symmetric monoidal smash product of spectra are due to
The monoidal model structure on is discussed for instance in
Relation to symmetric monoidal (infinity,1)-categories is discussed in